\newproblem{lay:6_1_24}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 6.1.24}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Carlos Oscar Sorzano, Aug. 31st, 2013} \\}{}

  % Problem statement
	Verify the \textit{Parallelogram Law} for vectors $\mathbf{u}$ and $\mathbf{v}$ in $\mathbb{R}^n$:
	\begin{center}
		$\|\mathbf{u}+\mathbf{v}\|^2+\|\mathbf{u}-\mathbf{v}\|^2=2\|\mathbf{u}\|^2+2\|\mathbf{v}\|^2$
	\end{center}
}{
   % Solution
	We know that $\|\mathbf{x}\|^2=\mathbf{x}\cdot\mathbf{x}$. In $\mathbb{R}^n$ the most standard inner product is defined as
	\begin{center}
		$\mathbf{x}\cdot \mathbf{y}=\mathbf{x}^T\mathbf{y}$
	\end{center}
	Then,
	\begin{center}
		$\begin{array}{rcl}
			\|\mathbf{u}+\mathbf{v}\|^2+\|\mathbf{u}-\mathbf{v}\|^2&=&(\mathbf{u}+\mathbf{v})^T(\mathbf{u}+\mathbf{v})+(\mathbf{u}-\mathbf{v})^T(\mathbf{u}-\mathbf{v}) \\
			   &=&(\|\mathbf{u}\|^2+\|\mathbf{v}\|^2+2\mathbf{u}\cdot\mathbf{v})+(\|\mathbf{u}\|^2+\|\mathbf{v}\|^2-2\mathbf{u}\cdot\mathbf{v})\\
				 &=&2\|\mathbf{u}\|^2+2\|\mathbf{v}\|^2
		 \end{array}$
	\end{center}
	
}
\useproblem{lay:6_1_24}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}
